Singular Sturmian separation theorems on unbounded intervals for linear Hamiltonian systems

In this paper we develop new fundamental results in the Sturmian theory for nonoscillatory linear Hamiltonian systems on an unbounded interval. We introduce a new concept of a multiplicity of a focal point at infinity for conjoined bases and, based on this notion, we prove singular Sturmian separation theorems on an unbounded interval. The main results are formulated in terms of the (minimal) principal solutions at both endpoints of the considered interval, and include exact formulas as well as optimal estimates for the numbers of proper focal points of one or two conjoined bases. As a natural tool we use the comparative index, which was recently implemented into the theory of linear Hamiltonian systems by the authors and independently by J. Elyseeva. Throughout the paper we do not assume any controllability condition on the system. Our results turn out to be new even in the completely controllable case.     

On the number of an eigenvalue of a general nonlinear self-adjoint spectral problem for systems of ordinary differential equations

In the case of a general nonlinear self-adjoint spectral problem for systems of ordinary differential equations with boundary conditions independent of the spectral parameter, we introduce the notion of the number of an eigenvalue. Methods for the computation of the numbers of eigenvalues lying in a given range of the spectral parameter and for finding the eigenvalue with a given number, which were earlier suggested by the author for Hamiltonian systems, are generalized to the considered problem. We introduce the notion of an index of a problem for a general nontrivially solvable linear homogeneous self-adjoint boundary value problem.     

Multiplicities of focal points for discrete symplectic systems: revisited

In this note, we define a notion of multiplicity of focal points for conjoined bases of discrete symplectic systems. We show that this definition is equivalent to the one given by Kratz in [Discrete oscillation, J. Difference Equ. Appl., 9(1), 135–147 (2003)] and, furthermore, it has a natural connection to the newly developed continuous time theory on linear Hamiltonian differential systems. Many results obtained recently by Bohner, Do?lý, and Kratz regarding the non-negativity of the corresponding discrete quadratic functionals, Sturmian separation and comparison theorems, and oscillation theorems relating the number of focal points of a certain special conjoined basis with the number of eigenvalues of the associated discrete symplectic eigenvalue problem, are now formulated in terms of this alternative definition of multiplicities.     

Comparison theorems for symplectic systems of difference equations

In the present paper, we prove comparison theorems for symplectic systems of difference equations, which generalize difference analogs of canonical systems of differential equations. We obtain general relations between the number of focal points of conjoined bases of two symplectic systems with matrices W _i and $ \hat W_i $ \hat W_i as well as their corollaries, which generalize well-known comparison theorems for Hamiltonian difference systems. We consider applications of comparison theorems to spectral theory and in the theory of transformations. We obtain a formula for the number of eigenvalues λ of a symplectic boundary value problem on the interval (λ ₁, λ ₂]. For an arbitrary symplectic transformation, we prove a relationship between the numbers of focal points of the conjoined bases of the original and transformed systems. In the case of a constant transformation, we prove a theorem that generalizes the well-known reciprocity principle for discrete Hamiltonian systems.     

Nonlinear spectral problem for a self-adjoint vector differential equation

We consider a spectral problem that is nonlinear in the spectral parameter for a self-adjoint vector differential equation of order 2n. The boundary conditions depend on the spectral parameter and are self-adjoint as well. Under some conditions of monotonicity of the input data with respect to the spectral parameter, we present a method for counting the eigenvalues of the problem in a given interval. If the boundary conditions are independent of the spectral parameter, then we define the notion of number of an eigenvalue and give a method for computing this number as well as the set of numbers of all eigenvalues in a given interval. For an equation considered on an unbounded interval, under some additional assumptions, we present a method for approximating the original singular problem by a problem on a finite interval.     

Bifurcation of critical points for continuous families of C 2 functionals of Fredholm type

Given a continuous family of C ² functionals of Fredholm type, we show that the nonvanishing of the spectral flow for the family of Hessians along a known (trivial) branch of critical points not only entails bifurcation of nontrivial critical points but also allows to estimate the number of bifurcation points along the branch. We use this result for several parameter bifurcation, estimating the number of connected components of the complement of the set of bifurcation points in the parameter space and apply our results to bifurcation of periodic orbits of Hamiltonian systems. By means of a comparison principle for the spectral flow, we obtain lower bounds for the number of bifurcation points of periodic orbits on a given interval in terms of the coefficients of the linearization.     

Coupled Intervals in the Discrete Optimal Control

In this paper, we investigate the nonnegativity and positivity of a quadratic functional ? with variable (i.e. separable and jointly varying) endpoints in the discrete optimal control setting. We introduce a coupled interval notion, which generalizes (i) the conjugate interval notion known for the fixed right endpoint case and (ii) the coupled interval notion known in the discrete calculus of variations. We prove necessary and sufficient conditions for the nonnegativity and positivity of ? in terms of the nonexistence of such coupled intervals. Furthermore, we characterize the nonnegativity of ? in terms of the (previously known notions of) conjugate intervals, a conjoined basis of the associated linear Hamiltonian system, and the solvability of an implicit Riccati equation. This completes the results for the nonnegativity that are parallel to the known ones on the positivity of ?. Finally, we define partial quadratic functionals associated with ? and a (strong) regularity of ?, which we relate to the positivity and nonnegativity of ?.     

A $${C^\infty}$$ closing lemma for Hamiltonian diffeomorphisms of closed surfaces

We prove a \({C^\infty}\) closing lemma for Hamiltonian diffeomorphisms of closed surfaces. This is a consequence of a \({C^\infty}\) closing lemma for Reeb flows on closed contact three-manifolds, which was recently proved as an application of spectral invariants in embedded contact homology. A key new ingredient of this paper is an analysis of an area-preserving map near its fixed point, which is based on some classical results in Hamiltonian dynamics: existence of KAM invariant circles for elliptic fixed points, and convergence of the Birkhoff normal form for hyperbolic fixed points.     

Dynamics of regularized cavity flow at high Reynolds numbers

A numerical simulation of the unsteady incompressible flows in the unit cavity is performed by using a Chebychev-Tau approximation for the space variables. The high accuracy of the spectral methods and the condensed distribution of the Chebychev-collocation points near the boundary enable us to obtain reliable results for high Reynolds numbers with a moderate number of modes. It is found that a stationary solution always exists for Reynolds numbers up to 10000. For Reynolds numbers larger than a critical value which is between 10000 and 12000, no more steady solution is found, instead, there is a persistent oscillation indicating a Hopf bifurcation.     

Every Banach algebra has the spectral radius property

Nylen and Rodman [NR] introduced the notion of spectral radius property in Banach algebras in order to generalize a classical theorem of Yamamoto on the asymptotic behaviour of the singular values of ann xn matrix. In this paper we prove a conjecture of theirs in the affirmative, namely that any unital Banach algebra has the spectral radius property. In fact a slightly more general spectral property holds. We show that for every element which has spectral points which are not of finite multiplicity, the essential spectral radius is the supremum of the set of absolute values of the spectral points that are not of finite multiplicity.     

Multiparameter spectral problem for some weakly coupled systems of Hamiltonian equations

We consider a multiparameter spectral problem for a weakly coupled system of ordinary differential equations in which every equation is Hamiltonian and contains two unknown functions. Using the notion of the number of an eigenvalue for a problem with one such equation, we give a statement of the problem of finding the desired eigentuple of values for the problem with several equations. We prove the existence and uniqueness of a solution of this problem and suggest and study a numerical solution method.     

On the self-adjoint nonlinear eigenvalue problem for Hamiltonian systems of ordinary differential equations with singularities

Certain properties of the nonlinear self-adjoint eigenvalue problem for Hamiltonian systems of ordinary differential equations with singularities are examined. Under certain assumptions on the way in which the matrix of the system and the matrix specifying the boundary condition at a regular point depend on the spectral parameter, a numerical method is proposed for determining the number of eigenvalues lying on a prescribed interval of the spectral parameter.     

Small Spectral Gap in the Combinatorial Laplacian Implies Hamiltonian

We consider the spectral and algorithmic aspects of the problem of finding a Hamiltonian cycle in a graph. We show that a sufficient condition for a graph being Hamiltonian is that the nontrivial eigenvalues of the combinatorial Laplacian are sufficiently close to the average degree of the graph. An algorithm is given for the problem of finding a Hamiltonian cycle in graphs with bounded spectral gaps which has complexity of order n ^{cln n} .     

Symplectic Geometry of Manifolds with Almost Nowhere Vanishing Closed 2-Form

We study local geometric properties of manifolds equipped with a closed 2-form nondegenerate at all points of a dense proper subset. We introduce the natural notion of tame singular point, at which the matrix of the 2-form degenerates in a regular way. We find a condition for Hamiltonian dynamical systems to be extended smoothly to tame singular points, generalize the Darboux theorem about the local reduction of the matrix of the 2-form to canonical form, and study the singular behavior of directional gradients.     

Determination of the number of an eigenvalue of a singular nonlinear self-adjoint spectral problem for a linear Hamiltonian system of differential equations

We suggest a method for determining the number of an eigenvalue of a self-adjoint spectral problem nonlinear with respect to the spectral parameter, for some class of Hamiltonian systems of ordinary differential equations on the half-line. The standard boundary conditions are posed at zero, and the solution boundedness condition is posed at infinity. We assume that the matrix of the system is monotone with respect to the spectral parameter. The number of an eigenvalue is determined by the properties of the corresponding nontrivially solvable homogeneous boundary value problem. For the considered class of systems, it becomes possible to compute the numbers of eigenvalues lying in a given range of the spectral parameter without finding the eigenvalues themselves.     

Computing near-best fixed pole rational interpolants

We study rational interpolation formulas on the interval [−1,1] for a given set of real or complex conjugate poles outside this interval. Interpolation points which are near-best in a Chebyshev sense were derived in earlier work. The present paper discusses several computation aspects of the interpolation points and the corresponding interpolants. We also study a related set of points (that includes the end points), which is more suitable for applications in rational spectral methods. Some examples are given at the end of this paper.     

Relative oscillation of linear Hamiltonian differential systems

We discuss the concept of relative oscillation for linear Hamiltonian differential systems. We prove a statement which enables to study this oscillation using the criteria of the classical oscillation theory. We also discuss some other aspects of the problem.     

Interval oscillation criteria for linear Hamiltonian systems

In this paper some new oscillation criteria of interval type for linear matrix Hamiltonian systems are established which are different from most known ones in the sense that they are based on the information only on a sequence of subinterval of [t₀, ∞) rather than on the whole half line. Our results are shaper than previous results. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

哈密顿系统的稳定性边界

通过采用摄动法对线性哈密顿参数系统的特征值和特征向量进行灵敏度分析,给出了此类系统的稳定性边界的判据,结果表明:具有约当链的系统重特征根对系统的稳定性起至关重要的作用。